A Sufficient Condition for the Liveness of Weighted Event Graphs

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A Sufficient Condition for the Liveness of Weighted Event Graphs

Olivier Marchetti, Alix Munier-Kordon

To cite this version:

Olivier Marchetti, Alix Munier-Kordon. A Sufficient Condition for the Liveness of Weighted Event

Graphs. [Research Report] lip6.2005.004, LIP6. 2005. �hal-02545677�

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p = (t i , t j ) ∈ P

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p

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w(p) v(p)

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w(p) > M ₀ ^∗ (p) + w(p)ν _i − v(p)ν _j

zòuî 0ìêðëføGç

M ₀ ^∗ (p) + w(p)ν _i − v(p)ν _j ≡ 0(gcd _p )

max(w(p) − v(p), 0) ≡ 0(gcd _p )

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>îiç(÷›çJè -

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<iïÎçuíéïðûb

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zòuî fìÂêðëføGç

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^{‘îºç(÷›çJè}

w(p) − gcd p ≥ M ₀ ^∗ (p) + w(p)ν i − v(p)ν j

zì

R gcd (M ₀ (p)) < gcd p

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< t i , ν i >

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< t j , ν j >

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∆.w(p ₁ ) > ∆.[M ₀ (p ₁ ) + w(p ₁ ).ν i − v(p ₁ ).ν j ] > ∆. max(w(p ₁ ) − v(p ₁ ), 0) m

w(p ₂ ) > M ₀ (p ₂ ) + w(p ₂ ).ν _i − v(p ₂ ).ν _j > max(w(p ₂ ) − v(p ₂ ), 0)

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a ∈ B

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(N ₁ , ..., N n ) ∈ { N ^∗ } ⁿ

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N = lcm i∈{1,...,n} (N i )

λ = lcm a∈{1,...,m} (w(p a ), v(p a ))

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p a = (t i , t j )

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α a = Z i

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Z _i

α _a ∈ N ^?

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p _a = (t _j , t _i )

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N i

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w(p a ) v(p a ) = α a v(p a )

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(p a , p b ) ∈ P ⁺ (t i ) × P ⁺ (t i )

α _a w(p _a ) = Z _i = α _b w(p _b )

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(p a , p _b ) ∈ P ⁻ (t i ) × P ⁻ (t i )

^î,êðè

p a = (t j , t i )

α _a v(p _a ) = Z j

w(p _a ) v(p _a ) = N λ N _j

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α a v(p a ) = Z i = α b v(p b )

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Z i

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Z = lcm i∈{1,...,n} (Z _i )

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N _i = _Z ^Z

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p a = (t i , t j ) ∈ P

N _i

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∀i ∈ {1, ..., n}

N i .Z i = K

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(Z i ) _i∈{1,..,n}

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(N i ) _i∈{1,..,n}

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∀i ∈ {1, ..., 5}

N _i .Z _i = 2520

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t N = (84, 70, 35, 70, 40)

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2 18 15

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21

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t 5

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G

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ν _i

i ∈ {1, ..., q}

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t i

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p q = (t q , t ₁ )

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X q

i=1

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v(p _i ) = w(p _i+1 )

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P q

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C _Dl

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∀p ∈ C _Dl ∩ P, M (p) < v(p)

nìèfçëm‘ô çJé(òóÉèÂòÅñ›çJëfìnù‘éÂçGìçJë«èêðëdíhù‘ï=íÅøGç

p

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∀p ∈ C _Dl ∩ P, M(p) ≤ v(p) − gcd _p

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X

p∈C Dl ∩P

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p∈C Dl ∩P

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M ₀ (p)

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t ₁

p ₁

M 0 (p 1 )

w(p ₁ ) v(p ₁ )

t ₂ p ₂

M 0 (p 2 ) w(p ₂ ) = v(p ₁ ) v(p ₂ ) = w(p ₁ )

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M ₀ (p ₁ ) + M ₀ (p ₂ ) > v(p ₁ ) + v(p ₂ ) − 2.gcd p ₁

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w(p ₁ ) = v(p ₂ )

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gcd _{p 1} = gcd _{p 2} = w

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p 1

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M 0 (p 1 ) w

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M 0 (p 2 ) w

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M ₀ (p ₁ )

w + M ₀ (p ₂ ) w > 0

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M ₀ (p ₁ )

^íëfö

M ₀ (p ₂ )

^íéÂç(ô ‘ïðèêðù‘ïÎçWòó

w

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w(p ₁ ) 6= w(p ₂ )

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p = p ₁

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(M (p ₁ ), M (p ₂ ))

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ν 1

íëfö

ν 2

ìfø/%è>íè -

M (p ₁ ) + ν ₁ .w(p) − ν ₂ .v(p) = M(p ₁ ) = ⇒ ν ₁ .w(p) = ν ₂ .v(p) M (p ₂ ) + ν ₂ .v(p) − ν ₁ .w(p) = M(p ₂ ) = ⇒ ν ₂ .v(p) = ν ₁ .w(p)

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ν ₁ ^∗ = ^lcm _w(p) ^p

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ν ₂ ^∗ =

lcm p

v(p)

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Λ

êÎìò,‘ëfö‘çGö;«û -

Λ ≥ lcm _p

w(p) + lcm _p v(p) − 1

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σ = w(p) + v(p) − 2.gcd p

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p

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gcd p

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| {z }

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M (p ₁ ) = v(p) − gcd p

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M(p ₂ ) = v(p ₂ ) − gcd _p = w(p) − gcd _p

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σ

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p ₁ p ₃

p ₂

p ₄ 3

3 t ₁

2

t ₄ 2

2 2 t 2 3

t 3 3

0

0

4 1

¸êð÷‘éÂçxý“ý-

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t ₃ 2.t ₄ t ₁ t ₂ t ₃ t ₄ t ₁ 2.t ₂

M(p ₁ ) : 0 0 0 3 1 1 1 4 0

M(p 2 ) : 4 1 1 1 3 0 0 0 4

M(p ₃ ) : 1 4 0 0 0 3 1 1 1

M(p ₄ ) : 0 0 4 1 1 1 3 0 0

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